L(s) = 1 | + 3-s + 7-s + 9-s − 6·13-s + 6·17-s + 21-s − 23-s + 27-s + 6·29-s − 8·31-s − 10·37-s − 6·39-s − 6·41-s − 8·43-s + 49-s + 6·51-s − 2·53-s − 4·59-s + 10·61-s + 63-s + 8·67-s − 69-s − 10·73-s + 16·79-s + 81-s − 8·83-s + 6·87-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 0.377·7-s + 1/3·9-s − 1.66·13-s + 1.45·17-s + 0.218·21-s − 0.208·23-s + 0.192·27-s + 1.11·29-s − 1.43·31-s − 1.64·37-s − 0.960·39-s − 0.937·41-s − 1.21·43-s + 1/7·49-s + 0.840·51-s − 0.274·53-s − 0.520·59-s + 1.28·61-s + 0.125·63-s + 0.977·67-s − 0.120·69-s − 1.17·73-s + 1.80·79-s + 1/9·81-s − 0.878·83-s + 0.643·87-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 193200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 193200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 + T \) |
good | 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 + 8 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.34410896391823, −12.75921006378307, −12.43326539524316, −11.87219123196206, −11.73110627093034, −10.90534483792361, −10.24121561680017, −10.11873964259615, −9.625521914165237, −9.065493231333345, −8.481426202639901, −8.142122981978116, −7.578657711224195, −7.130517454564748, −6.836193701564328, −6.006568048584970, −5.377090331180987, −4.974235045286887, −4.635641556059824, −3.699113353967548, −3.387720305646491, −2.817195260910187, −2.024632509062717, −1.752674327921967, −0.8403447579263040, 0,
0.8403447579263040, 1.752674327921967, 2.024632509062717, 2.817195260910187, 3.387720305646491, 3.699113353967548, 4.635641556059824, 4.974235045286887, 5.377090331180987, 6.006568048584970, 6.836193701564328, 7.130517454564748, 7.578657711224195, 8.142122981978116, 8.481426202639901, 9.065493231333345, 9.625521914165237, 10.11873964259615, 10.24121561680017, 10.90534483792361, 11.73110627093034, 11.87219123196206, 12.43326539524316, 12.75921006378307, 13.34410896391823